Integrand size = 25, antiderivative size = 95 \[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{5/2} \, dx=-\frac {c (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{3 b d}+\frac {c^2 \sqrt {d \cos (a+b x)} E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sin (a+b x)}}{2 b \sqrt {\sin (2 a+2 b x)}} \]
-1/3*c*(d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^(3/2)/b/d-1/2*c^2*(sin(a+1/4*Pi +b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticE(cos(a+1/4*Pi+b*x),2^(1/2))*(d*c os(b*x+a))^(1/2)*(c*sin(b*x+a))^(1/2)/b/sin(2*b*x+2*a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.71 \[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{5/2} \, dx=\frac {2 \sqrt {d \cos (a+b x)} \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {7}{4},\frac {11}{4},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{5/2} \tan (a+b x)}{7 b} \]
(2*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 7/4, 11/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(5/2)*Tan[a + b*x])/(7*b)
Time = 0.40 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3048, 3042, 3052, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (c \sin (a+b x))^{5/2} \sqrt {d \cos (a+b x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c \sin (a+b x))^{5/2} \sqrt {d \cos (a+b x)}dx\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {1}{2} c^2 \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}dx-\frac {c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{3 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} c^2 \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}dx-\frac {c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{3 b d}\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle \frac {c^2 \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{2 \sqrt {\sin (2 a+2 b x)}}-\frac {c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{3 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {c^2 \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{2 \sqrt {\sin (2 a+2 b x)}}-\frac {c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{3 b d}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {c^2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{2 b \sqrt {\sin (2 a+2 b x)}}-\frac {c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{3 b d}\) |
-1/3*(c*(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2))/(b*d) + (c^2*Sqrt[d *Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(2*b*Sqr t[Sin[2*a + 2*b*x]])
3.3.78.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(409\) vs. \(2(106)=212\).
Time = 0.26 (sec) , antiderivative size = 410, normalized size of antiderivative = 4.32
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {d \cos \left (b x +a \right )}\, \sqrt {c \sin \left (b x +a \right )}\, \left (2 \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )-6 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+3 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-6 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+3 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-5 \sqrt {2}\, \left (\cos ^{2}\left (b x +a \right )\right )+3 \sqrt {2}\, \cos \left (b x +a \right )\right ) c^{2} \sec \left (b x +a \right ) \csc \left (b x +a \right )}{12 b}\) | \(410\) |
1/12/b*2^(1/2)*(d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(1/2)*(2*2^(1/2)*cos(b* x+a)^4-6*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)* (cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticE((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1 /2*2^(1/2))*cos(b*x+a)+3*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc( b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticF((-cot(b*x+a)+csc(b *x+a)+1)^(1/2),1/2*2^(1/2))*cos(b*x+a)-6*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)* (cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticE((- cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))+3*(-cot(b*x+a)+csc(b*x+a)+1)^( 1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*Ellipti cF((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))-5*2^(1/2)*cos(b*x+a)^2+3* 2^(1/2)*cos(b*x+a))*c^2*sec(b*x+a)*csc(b*x+a)
\[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{5/2} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{5/2} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
\[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{5/2} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{5/2} \, dx=\int \sqrt {d\,\cos \left (a+b\,x\right )}\,{\left (c\,\sin \left (a+b\,x\right )\right )}^{5/2} \,d x \]